The mixing vanes generate vortices, which are a key factor driving the subchannel crossflow. Therefore, the formation of the vortices, as well as their path through the rod bundle, is investigated. First, the formation of the vortices at the mixing vanes is illustrated. To identify a vortex in the flow, the Q-criterion can be used. The Q-criterion estimates the regions in the flow field that are dominated by rotation, as well as the regions dominated by strain and shear, and is calculated by (Chang and Tavoularis, 2015) Q = − 1 2 S i j S i j − Ω i j Ω i j , (14) S i j = 1 2 ∂ u i ∂ x j + ∂ u j ∂ x i , (15) Ω i j = 1 2 ∂ u i ∂ x j − ∂ u j ∂ x i . (16)

A positive Q-value indicates a region that is dominated by rotation, while a negative Q-value indicates a region that is dominated by strain and shear. By drawing iso-value surfaces with a certain positive value, vortex envelopes can be made visible. Figure 4 shows the iso-surface of Q = 8.5 * 10 5 s − 2 near mixing vanes. The two main vortices in each subchannel are formed at the root of the mixing vanes. For clarity, the iso-surface with a high value of Q is chosen to show the two strong vortices. On the tips, edges, and knees of mixing vanes and strips of the space grid, weak shedding vortices are formed.

Figure 5 shows the Q-criterion in subchannel No. 11. The regions with a high Q-value are indications of vortices. At z = 0.25 D_h, the two main vortices can be seen, located at the root edges of mixing vanes close to the subchannel center. The secondary vortices generated by the edges of mixing vanes in gaps can be seen. From z = 0.25 D_h to 1 D_h, the two main vortices rotate around the subchannel center in a clockwise direction. In addition, the rotating vortices are instantaneously temporal. Within 1 D_h, the secondary vortices formed at the sharp edges of the adjacent subchannels move into subchannel 11. At z = 0.75 D_h, the generation of the secondary vortices at the tips of the mixing vanes can be seen. On the way upstream, these secondary vortices move to the adjacent subchannel at the top and bottom. At z = 1.25 D_h, the secondary vortices induced by the top edges of the mixing vanes in subchannels 10 and 12 (refer to Figure 1B) enter subchannel 11. Furthermore, these vortices move toward the subchannel center, while the main vortices move toward the adjacent subchannels. There is still a little clockwise spinning in the path of the main vortices. The secondary vortices generated by the sharp edges move into the adjacent subchannels. At z = 2.75 D_h, the down main vortex has already moved into subchannel 15 and the top main vortex moves into subchannel 7. Moreover, the main vortices from subchannels 10 and 12 enter subchannel 11. The secondary vortices of the top edges of the mixing vanes in the adjacent subchannels decompose, while the main vortices from the adjacent subchannels move toward the subchannel center along a clockwise path close to the rods. Another important effect of heat transfer from the fuel rods to the coolant flow is the breaking up of the boundary layer around the fuel rods. Generally, in a pipe flow, the friction of the wall leads to a shear-dominated boundary layer near the walls. However, in Figure 5, the Q-criterion indicates regions with a high rotation close to rods. These are the regions where the boundary layer is broken up by flowing vortices.

Figure 6A shows the vector field of the crossflow velocity near the rod at the cross section at z = 1 D_h downstream of the spacer grid. Figure 6B shows the Q-criterion at the same location. Figure 6B shows a region of high Q-value in the boundary layer close to the rod (marked red). Figure 6A shows the reason for this effect. The vortex induces crossflow velocity in such a way that the boundary layer is very thin at this position. The wall-normal velocity gradient is very high at this location. Therefore, the heat transfer from the fuel rod into the coolant flow is enhanced at this location. The rest of the section of the rod wall is surrounded by a normal boundary layer, and there is no high wall-normal velocity gradient. In this part of the section of the wall, the Q-value is low, and therefore, the flow is shear-dominated. The cross-flow velocity and Q-criterion at z = 3.75 D_h are shown in Figure 6C, D, respectively. The regions of high Q-value in Figure 6D are marked in red, and the corresponding regions with a thin boundary layer are marked in red in Figure 6C. A region with a relatively thick boundary layer is marked in blue. Here, the Q-criterion has a small value.

Figure 7 shows U R M S at z = 1 D_h, 4 D_h, 8 D_h, and 16 D_h in the central four subchannels. The mixing vanes induce a strong shear flow through gaps at z = 1 D_h. When the shear flow sweeps rods, high U R M S is generated near the rods. Generally, high U R M S is always located in a vortex or near the rods. At z = 4 D_h, the maxima U R M S is stronger in the subchannel center, resulting in a chessboard pattern in the cross section. The new vortices merge with vortices from adjacent subchannels, while some vortices decompose and transfer their kinetic energy into the merging vortex. This process exists in subchannel centers, resulting in high U R M S in subchannel centers. At z = 8 D_h, the chessboard pattern maintains, but strongest U R M S is in gaps and near rods. At z = 16 D_h, the chessboard pattern is weak, while strongest U R M S still locates in gaps. V R M S is shown in Figure 8, at z = 1 D_h, 4 D_h, 8 D_h, and 16 D_h downstream of the spacer grid. Generally, the appearance of the fluctuations of the velocity component v is like the appearance of the fluctuations of the velocity component u , but the distribution is vice versa. The maxima of the fluctuations can be found in the vertical gaps because the flow can pass through these gaps with the v component of the velocity. Moreover, the chessboard pattern is appearing from z = 4 D_h to z = 16 D_h, while the chessboard pattern and magnitude of V R M S fade with the flow development downstream of the spacer grid. The decay rates of U R M S and V R M S are high in the near wake of the spacer grid within 4 D_h, where the Reynolds shear stress is strong. After 4 D_h, the decay rates of U R M S and V R M S become slow.

The lateral turbulence intensity can be calculated by T I = 1 w ¯ b 1 2 U R M S 2 + V R M S 2 , (17) u ′ v ′ ¯ = 1 N − 1 ∑ i = 1 N u i − U v i − V . (18)

Figure 9 shows the lateral turbulence intensity at 1 D_h, 4 D_h, 8 D_h, and 16 D_h downstream of the spacer grid. At z = 1 D_h, the turbulence intensity has the maxima close to the rods, near the gaps of the subchannels. The high turbulence intensity is at the same locations where the velocity maxima are appearing due to the vortices. In the subchannel center, the turbulence intensity is very small. At z = 4 D_h, the maxima of the turbulence intensity are in the subchannel centers, while the magnitude of the maxima and the averaged turbulence intensity are smaller than those at the first cross section. The shape of the distribution of the turbulence intensity is very similar to the shape of the distribution of the vortices, as observed by the Q-criterion. It is obvious that the vortices are leading to a high turbulence intensity. At z = 8 D_h, the decay of the turbulence intensity becomes slower. The maxima are moving from the subchannel centers toward the rods, especially into the gaps in between the adjacent subchannels. The high turbulence intensity close to the fuel rods is enhancing heat transfer at the locations in the gaps. At z = 16 D_h, the highest TI is located close to the rods in the vertical gaps, while next to the rods in the horizontal gaps, the smallest values of the turbulence intensity can be found. Therefore, the heat transfer is enhanced by the turbulence in the vertical gaps but not in the horizontal gaps. The turbulence intensity follows an exponential decay rate within 20 D_h downstream of the spacer grid.

The lateral Reynolds shear stress τ u v at z = 1 D_h is shown in Figure 10A. The counter-current crossflow in the subchannels near the spacer gird produces a strong shear flow through gaps into adjacent subchannels. The maxima of Reynolds shear stress τ u v can be seen in the gaps close to the rods, which correspond with the locations of a strong shear flow. τ u v from z = 4 D_h to 16 D_h can be seen in Figure 10B, D. In a streamwise direction, a strong decay of the magnitude of the shear flow is observed. At z = 4 D_h, the maximum τ u v is still in the gaps close to rods and vortices in the subchannel center. At z = 8 D_h and 16 D_h, the highest Reynolds stress is near rod surfaces, facing subchannel centers.

One key parameter to evaluate the effect of the spacer grid on the fluid flow is the area-averaged secondary flow intensity (SFI). The SFI is the area-averaged ratio of the magnitude of the spanwise crossflow velocity components u and v , and the streamwise component w , and therefore gives some indication of the inter-subchannel mixing. A strong secondary flow in the rod bundle leads to inter-subchannel mixing. The inter-subchannel mixing increases the heat transfer from the fuel rods into the coolant flow and is therefore very important for the investigation. Hence, it is important to validate the simulation for this parameter. The SFI can be calculated as (Busco and Hassan, 2018) S F I = 1 A ∑ i A i u ¯ 2 + v ¯ 2   w ¯ . (19)

In this equation, A represents the area of the cross section; A i is the measurement resolution or grid resolution, A i = ∆ x * ∆ y ; u and v are the crossflow velocity components in the respective directions; and w ¯ is the bulk flow velocity. The SFI follows an exponential function: S F I = S F I 0 + A e − B z D h , (20) which is evaluated from the measured data from 1 D_h to 20 D_h downstream of the spacer grid in the experiment and LES. The coefficients of fitted SFI correlations are listed in Table 1, and the fitted SFI correlations are listed in Figure 11.

In order to visualize the fully developed flow, far downstream of the spacer grid, the functions are plotted up to 40 D_h downstream of the spacer grid. It is important to be aware that drawing conclusions from extrapolated data is a challenge. Nevertheless, it can be assumed that the fitting function of the experiment gives a good approximation to the fully developed SFI₀, as well as to the axial distance downstream of the spacer grid, where the flow is fully developed. The fully developed secondary flow intensity SFI₀ is almost the same in the experiment and LES. Furthermore, the value of SFI₀, according to both fitting functions, almost reached approximately 30 D_h downstream of the spacer grid. Hence, it can be concluded that the accuracy of the LES approach is satisfactory for the SFI.